Impossible to snap to objects
You could try the following:

1: In Edit mode select the vertice you want to use as the snap point.

2: Move the 3D cursor to this vertice, (Shft S to open the snap menu, then choose  Cursor to Selected).

3: In Object mode move origin to the 3D cursor, (Shft Ctrl Alt C and choose the Origin to 3D Cursor option).

4: Change the Snap Target from Closest to Active.

In the example gif below, I’m trying to snap the point of the pyramid object to one of the corner vertices of the cube object.